A264447 a(n) = n*(n + 7)*(n + 14)*(n + 21)/24.

0, 110, 276, 510, 825, 1235, 1755, 2401, 3190, 4140, 5270, 6600, 8151, 9945, 12005, 14355, 17020, 20026, 23400, 27170, 31365, 36015, 41151, 46805, 53010, 59800, 67210, 75276, 84035, 93525, 103785, 114855, 126776, 139590, 153340, 168070, 183825, 200651, 218595, 237705, 258030

Offset: 0

Author

Peter Bala, Nov 13 2015

Keywords

• nonn
• easy

Comments

It is well-known, and easy to prove, that the product of 4 consecutive integers n*(n + 1)*(n + 2)*(n + 3) is divisible by 4!. It can be shown that the product of 4 integers in arithmetic progression n*(n + r)*(n + 2*r)*(n + 3*r) is divisible by 4! if and only if r is not divisible by 2 or 3 (see A007310 for these numbers). This is the case r = 7.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A007310, A264443, A264444, A264445, A264446, A264448, A264449, A264450.

Programs

Magma

[n*(n+7)*(n+14)*(n+21)/24: n in [0..40]]; // Vincenzo Librandi, Nov 16 2015

Maple

seq( n*(n + 7)*(n + 14)*(n + 21)/24, n = 0..40 );

Mathematica

Table[n (n + 7) (n + 14) (n + 21)/24, {n, 0, 40}] (* Vincenzo Librandi, Nov 16 2015 *)
Table[Times@@(n+7*Range[0,3])/24,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,110,276,510,825},50] (* Harvey P. Dale, May 01 2017 *)

PARI

vector(100, n, n--; n*(n+7)*(n+14)*(n+21)/24) \\ Altug Alkan, Nov 15 2015

Formula

O.g.f.: x*(110 - 274*x + 230*x^2 - 65*x^3)/(1 - x)^5.

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4. - Vincenzo Librandi, Nov 16 2015